Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Zhou, Weien; Zhang, Jingjing; Hong, Jialin; Song, Songhe

doi:10.1016/j.cam.2017.04.050

Cited by 22 publications

(15 citation statements)

References 32 publications

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“…the Hamiltonian considered in [9] (where the matrix is diagonal), the linear stochastic oscillator from [44], and various stochastic Hamiltonian systems studied in [36,Chap. 4], see also [35], or [26,27,42,50].…”

Section: Settingmentioning

confidence: 99%

Drift-preserving numerical integrators for stochastic Poisson systems

Cohen

Vilmart

2021

International Journal of Computer Mathematics

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We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.

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Section: Settingmentioning

confidence: 99%

Drift-preserving numerical integrators for stochastic Poisson systems

Cohen

Vilmart

2021

International Journal of Computer Mathematics

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“…However, most SDEs arising in practice are nonlinear, and cannot be solved explicitly. There has been tremendous interests in developing effective and reliable numerical methods for SDEs during the last few decades, for example see [4][5][6][7][8][9][10][11][12][13][14]. Runge-Kutta (RK) methods with continuous stage were firstly presented by Butcher in 1970s [15], and they have been investigated and discussed by several authors recently because of the great advantages in conserving symplecticity [16], preserving energy [17] and so on.…”

Section: Introductionmentioning

confidence: 99%

Continuous stage stochastic Runge–Kutta methods

2021

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In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

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“…Stochastic Hamiltonian systems, as a kind of structure-preserving stochastic differential equations, are used to simulate dynamic systems under stochastic dissipative disturbance [25][26][27][28][29][30]. If the disturbances are seen as white noise, stochastic Hamiltonian systems can be written as stochastic differential equations driven by Wiener processes.…”

Section: Introductionmentioning

confidence: 99%

Symplectic-Structure-Preserving Uncertain Differential Equations

Yin

Gao

et al. 2021

Symmetry

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Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm.

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Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Cited by 22 publications

References 32 publications

Drift-preserving numerical integrators for stochastic Poisson systems

Drift-preserving numerical integrators for stochastic Poisson systems

Continuous stage stochastic Runge–Kutta methods

Symplectic-Structure-Preserving Uncertain Differential Equations

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